Method of measurement and apparatus therefor

ABSTRACT

Apparatus and method for determining a parameter which will effectively characterize the fluctuations of a dependent variable with respect to an independent variable, such as the height of a surface profile with respect to position along the surface. The apparatus includes means for detecting the variations of the independent variable such as a transducer for detecting the variations of the height of the surface profile along a line in the surface to obtain a signal representing the said variations, and means for differentiating the signal representing the said variations to provide a rate signal depending on the rate of change of the profile height, and means for combining the two signals, such as by integrating each and then dividing one by the other, to provide the said parameter which characterizes the fluctuations.

United States Patent 1 1 Spragg et al.

1 1March 13, 1973 1 1 METHOD OF MEASUREMENT AND 3,544,744 12 1970Peklenik ..73 105 x APPARATUS HEREFOR 3,313,149 4/1967 Spra ..73/105[75] Inventors: Robert Claude Spragg; David John Primary ExaminerMalcolm A. Morrison wh'telmuse both of Lelcester AssistantExaminer-Edward J. Wise gland Attorney-l-lolcombe, Wetherill & Brisebois[73] Assignee: The Rank Organization Limited,

London, England ABSTRACT [22] Filed: Jan. 29, 1971 Apparatus and methodfor determining a parameter which will effectively characterize thefluctuations of a [21] Appl' L070 dependent variable with respect to anindependent variable, such as the height of a surface profile with [30]Foreign Application Priority Data respect to position along the surface.The apparatus includes means for detecting the variations of the in-Jan. 30, 1970 Great Britain .44,58l/70 dependent variable Such as atransducer for detecting the variations of the height of the surfaceprofile along [52] 73/105 233/15 a line in the surface to obtain asignal representing-the [51] Int. Cl. ..G b 7/ Said variations and meansfor differentiating the signal [58] held of Search "235/ 15 I 2 21 lrepresenting the said variations to provide a rate signal 235/15 73/10532 /77 A depending on the rate of change of the profile height, andmeans for combining the two signals, such as by [56] References C'tedintegrating each and then dividing one by the other, to UNITED STATESPATENTS provide the said parameter which characterizes the fluctuations.3,112,642 l2/l963 Harmon et al ..73/ 1 3,123,999 3/1964 Judd i ..73/10552 Claims, 8 Drawing Figures 3,580,062 5/1971 Perthen et al. ..73/105 6'arm/4? 056710470? 21 L-| PHASE ///6// 7601500627? AMPL/HA'P. JE/Vfi/f/VfP455 DA'TZCTQ? fiNZ-A. 2 1 5 6 LOW/ A 5 m I? [/1758 g SQUARE MW WTMAA70R /?C77/75Q A/VDSTOPE 5/6/Y/1L 1500M? LAW D/F/[Ff/VWATfl/Q PC77f71 Z4WEEK/R AND 570R l ;-fi

SW/TOV/NG L Q 4K @45J a 1 O/V/DER PATENTEUHAR131973 3 720, 1

SHEET 2 OF 5 METHOD OF MEASUREMENT AND APPARATUS THEREFOR This inventionrelates to methods of testing the interrelationship of a dependentvariable and an independent variable and apparatus therefor.

Such a method of testing is particularly applicable to the control orimprovement of manufacture.

When any investigation is carried out, the results are frequentlyobtained as a series of values of a dependent variable corresponding togiven values of an independent variable. They may be presented as agraphical display or a tabular presentation of the magnitude of thedependent variable for constant increments of the independent variable.For example, such a display might show the profile of a surface along aline in a chosen direction; alternatively the variation of thetemperature of a body with the passage of time might be shown. The dataobtained often show cyclic variations or fluctuate in a random manner.Commonly the data exhibit a combination of these two.

In order to make comparisons between different sets of data it isconvenient to subject them to some form of statistical analysis. Onetechnique which is frequently employed is to carry out an averagingprocess and to express the result as a mean value. In the investigationof the profile of a surface, a mean value which finds wide applicationis the center line average" which is the arithmetic mean of theamplitude of the profile ordinates when measured from a predeterminedreference line.

Although such a mean value may be readily derived using simpleinstrumentation and does find wide application for comparison of surfaceof a similar nature, it is subject to severe limitations from the pointof view of engineering control, since it is in no way dependent on therate of change of the dependent variable with respect to the independentvariable, being merely a function of the distribution of the magnitudeof the dependent variable.

For more complete characterization of the dependence of one variable onanother it is desirable to specify in some statistical way thedistribution of the normalized rate of change of the variable. It hasnow become generally recognized that the auto-correlation function ofthe variables provides such a criterion. The auto-correlation function4) (r) of the fuction f(t) is defined by the relationship Unfortunately,however this cannot readily be obtained from low cost instruments and itis necessary to search further for a parameter for providing the desiredcharacterization.

As a result of investigation into the nature of various surfaces, it hasbeen found that the normalized root mean square moment of the powerspectrum of the components of spatial frequency of the profile of asurface in respect of its frequency weight is given by the ratio of theroot mean square magnitude of the profile gradient to the root meansquare value of the height. Further it has been found that thenormalized root mean square moment of the power spectrum is givenapproximately by the ratio of other mean magnitudes of the profilegradient, such as the mean modulus magnitude to the corresponding meanheight value.

As is well known, the relation of the power spectrum to theautocorrelation function is one of inverse Fourier transforms.Furthermore, the root mean square moment of the power spectrum can beshown to be related to the autocorrelation function as the secondderivative of ('r) for 1= 0, that is, as expressing the initial decay ofthe function.

Expressed mathematically, where L is the assessment length, x is theindependent variable, that is distance in the direction of measurement,w is a measure of spatial frequency, f(x) is the profile height of thesurface normal to the direction of movement of the stylus:

t 21 0 l )]T-= N f (m 0;- %L (5% f(a:)) d:c

L (mu f: imn x if T 01 z 1 L df(a:)

J; H10) d2: L dz dz It can be shown mathematically that 1 1, 2 ill) maylf( )l 1 L d d 2 It has been found that these expressions provide theindices Arms and Au designated the root mean square wavelength index"and average wavelength index respectively, either of which represents,together with the corresponding mean magnitude, a consistentcharacterization of, in the case of surface texture measurement,roughness, waviness and error of form (features of surface topographyhaving different characteristic spatial frequency) components and in thecase of other interdependent variables in other fields of measurement, ameans of providing a simple characterization of spectra. Reference inthis specification to an average wavelength index" as herein definedwill be understood, therefore to relate to an index of the form:

Similarly, reference in this specification to a root mean squarewavelength index as herein defined will be understood to relate to anindex of the form Since the mean magnitude of a variable quantity andthe mean magnitude of the rate of variation of one variable with anothermay each be readily determined independently, a useful criterion fortesting and characterizing the relationships of interdependent variablescan be specified.

Accordingly the present invention provides a method of characterizingthe spectrum of a dependent variable with respect to an independentvariable comprising the steps of determining the value of the dependentvariable over a range of values of the independent variable, deriving afirst signal representing the mean magnitude of the dependent variableover the said range, deriving a second signal representing the meanmagnitude of the rate of change of the dependent variable with respectto the independent variable over the said range and modifying one of thesignals by means of the other to provide a parameter characterizing thesaid spectrum.

Preferably, one signal is divided by the other to derive the saidparameter.

In one embodiment the signal representing the mean magnitude ofthe rateof change of the dependent variable over the said range is derived froma signal representing the value of the rate of change of the dependentvariable which is derived by differentiation of a signal representingthe value of the dependent variable over the said range.

Preferably the parameter is the average wavelength index as hereindefined and the two signals are combined by dividing the signalrepresenting the mean magnitude of the rate of change of the dependentvariable over the said range into the signal representing the meanmagnitude of the dependent variable over the said range.

Particularly, the parameter to be determined characterizes the profileof a surface ofa body, the dependent variable being the distance of thesurface of the body from a reference line and the independent variablebeing distance along the surface from a predetermined point.

Preferably the signal representing the value of the dependent variableis obtained by passing a probe in contact with the surface along apredetermined line to obtain a signal representing the distance of thesurface from a reference line.

According to another .aspect of the invention apparatus forcharacterizing the spectrum ofa dependent variable with respect to anindependent variable comprises means for determining the value of thedependent variable over a range of values of the independent variable,means for providing a first signal representing the mean magnitude ofthe dependent variable over the said range, means for providing a secondsignal representing the mean magnitude of the rate of change of thedependent variable with respect to the independent variable over thesaid range, and means for modifying one of the signals by means of theother to provide a signal representing a parameter characterizing thesaid spectrum.

Various embodiments of the invention will now be more particularlydescribed, by way of example, with reference to the accompanyingdrawings, in which:

FIG. 1 is a block diagram illustrating the elements of an instrumentsystem for the measurement of the average wavelength index of theprofile of a surface;

FIG. 2 is a representation of a typical roughness signal produced bypart of the instrument of FIG. 1;

FIG. 3 is a representation of a signal produced by passing the signal ofFIG. 2 through a square law rectifi- FIG. 4 is a representation of asignal produced by passing the signal of FIG. 2 through adifferentiating circuit;

FIG. 5 is a representation of the signal produced by passing the signalof FIG. 2 through a full-wave rectifier;

FIG. 6 is a block diagram of an alternative embodiment of a system forthe measurement of the average wavelength index of the profile of asurface;

FIG. 7 is a block diagram of a further alternative embodiment of anaverage wavelength index determining instrument; and

FIG. 8 is a representation of a waveform used for the purpose ofillustrating differences between the embodiments of FIGS. 6 and 7.

Referring now to FIG. 1 of the drawings a stylus arm 1 bears a stylus incontact with a test surface S. The other arm of the stylus carries asoft ferro-magrietic armature which is movable so as to varydifferentially the inductances of two symmetrical coils constituting atransducer 2. The stylus mountand transducer are constrained fortranslational movement substantially parallel to the test surface, thestylus tip following the profile of the asperities on the surface. 7

Movements of the stylus tip normal to the surface create changes ofinductance in the transducer coils. The coils are connected in adifferential circuit arrangement and are fed with a sine wave signalfrom a carrier oscillator 3. The output signals from the coils depend onthe varying inductance and are combined to produce a carrier signal withbalanced or double modulation.

The signals are amplified by an amplifier 4 and demodulated in a phasesensitive detector 5 to produce a signal which is a direct analogue ofthe surface profile traced by the stylus tip. The profile signal at thisstage contains components due to offset (a mean DC level determined bythe absolute position of the stylus tip), error of form (departure ofthe surface from true form) and waviness (periodic undulations) as wellas roughness (the micro-texture it is desired to investigate). Theunwanted components are all substantially removed by passing the signalthrough'a high-pass filter 6 whose turn-over frequency is chosenappropriately.

The signal emerging from the high pass filter is represented in FIG. 2.It is an analogue of the surface roughness profile and consists only ofAC components which may be derived as Fourier components by performingtransformation on the roughness signal.

For reasons which will be explained later the profile signal is passedthrough a low-pass filter 7 having a turnover frequency approximately 6octaves above that of the high pass filter 6. It is then rectified in asquare law rectifier 8, integrated over a complete traverse of thestylus and stored in a memory device 9. The amplitude of the stored andintegrated signal may be displayed on an indicator 10, which, if it hasa square law charac teristic will indicate the root mean square of theprofile amplitude.

The roughness signal from the low pass filter 7 is also fed to adifferentiator circuit 11 and then to a square law rectifier 12. Thedifferentiated profile signal is illustrated in FIG. 4. The rectifiedsignal is passed to an integrator and store 13 and the resultingintegral may be displayed as a mean square or root mean square value byan indicator 14.

The integrated signals from stores 9 and 13 are fed respectively to thedivisor and divider inputs of a divider circuit 15 and the output fed toan indicator 16 having a square law characteristic and displayed as thespacing index of the surface profile.

A square root process is convenientlyincorporated if the divider circuitis of the logarithmic generators type comprising two input logarithmicgenerators, a subtrac tor circuit, and an output anti-logarithmicgenerator. The square root process is then simply incorporated byinserting a gain of k either in or after the subtractor circuitaccording to the well known relation:

MlN=antilog 1% (log M log N,)]

Incorporation of the square root process enables a linear characteristicdisplay device to be used.

A further advantage may be derived from such a logarithmic divider whichcan convert the scales for indicating root mean square amplitude androot mean square slope to a linear law. By means of switches, 17, thestored signals from integrators 9 and 13 may, when required, be switchedas divisors to the logarithmic divider circuit. As the circuit permitsthe derivation of square roots the divider input is simultaneouslyswitched to unit value for indication of either root mean squareamplitude or root mean square slope.

FIG. 6 is a block diagram of an instrument system for the measurement ofthe average wavelength of a roughness profile. This system closelyresembles that of FIG. 1 but differs in principle by the substitution ofmean modulus parameters for root mean square parameters. The system ofFIG. 6 produces a value of average wavelength close to that produced bythe system of FIG. 1 on account of the relation:

which is an approximate one but usefully accurate.

Themean modulus amplitude, results from integrating f(x) over theassessment traverse and adding ordinate values of f(x) irrespective ofwhether these ordinates are positive or negative with respect to a zerodatum. The square law rectifier 8 of FIG. 1 is replaced by a full waverectifier 18 in the arrangement shown in FIG. 6. The output waveform ofthe full wave rectifier circuit when fed with the profile signal of FIG.2 is illustrated in FIG. 5. The waveform of FIG. 5 shows the negativeportions of the waveform appearing inverted as positive portions and thesignal otherwise unchanged. A reversed inversion may be chosen.Comparison with the square law rectified waveform of FIG. 3 shows asimilar inversion of negative portions, but the square law effects thewaveform generally, large amplitude portions becoming peaky and lowamplitude portions compressed".

Likewise the mean modulus of the slope ldf(x,)/dx| may be employedinstead of the mean square by replacement of the square law rectifier,12,0f FIG. 1 by the full wave rectifier 19 in FIG. 6. For the system ofFIG. 6 neither square law scales nor the alternative of square rootextraction are required for readouts on the meters 10, 14, 16. Onlylinear scales and a straightforward dividing operation are required inthe design of items l0, l4, l6 and 15.

An alternative form of apparatus more compatible with digital techniquesmay be used to derive the. 1

average wavelength from the ratio of the mean amplitude of the profileand the mean modulus of the slope.

The slope of a line is defined as the differential of its amplitude and,to a first approximation is proportional to the number of predeterminedincrements Af(x) ot the amplitude per increment of distance Ax.

The apparatus shown in FIG. 7, in common with the embodiments-of FIGS. 1and 6 feeds a profile signal through a phase sensitive detector and lowand high pass filters. The mean amplitude is derived, as in theembodiment of FIG. 6 by means of a full-wave rectifier and an integratorand store 13. In this apparatus, however, increments of f(x) are set upas equally spaced voltage levels in reference level generator 20, eachlevel being associated with a crossings detector 22. The crossingsdetectors emittrigger pulses as intercepts of voltage levels areregistered and, as timeis proportional to distance,slope becomesrepresented as trigger pulses per unit time. In FIG. 8 reference levelsare su' perimposed on the plot of signal against time. When the signalslope is large, trigger pulses occur at a fast rate, and conversely, ata slow rate when the signal slope is small.

In order to obtain the mean slope the pulse rate must be integrated andstored, and this is conveniently effected by counting the trigger pulsesin counter 23.

For reasons of economy the reference voltage levels are not chosenarbitrarily. The instrument measures the parameter and in determiningthe scale range for this purpose, a related set of reference voltagelevels extending to approximately 3 are set up by means of the rangeswitch 21.

Divider '15 is of a hybrid analogue/digital form com-' prising a see-sawamplifier with input resistor R and has an output proportional to l/RGwhere G is the conductance of the network 24. This latter comprisesseparate weighting resistances controlled by digital switching from thecounter 23. The output of the divider amplifier is a measure of averagewavelength and is indicated by a meter 16 with linear scales.

The mean modulus of the amplitude may be indicated in the usual way by ameter 10 also with a linear scale. The average slope may be indicated bya meter,

14, which is conveniently switched to the amplifier and weightingresistance network reorganizedas a digital! analogue converter for thispurpose by switching internal connections. Alternatively the counter maybe. used.

. to drive a numerical readout of average slope.

From computation of the power spectrum and correlation functions of alarge number of practical machined surfaces, many surfaces may beclassified, into two categories; first order random and second orderrandom. For instance abraded and ground surfaces tend to have a whitenoise spectrum limited at the high frequencies by a drop off of about6dB/octave whereas surfaces produced by many single tooth machiningprocesses have a sharp peak in the spectrum on a broad random base whichtails off at high frequencies typical in fact of a second order randomprocess. The way in which this sort of surface differs from the periodicis in the shape of the amplitude distribution. For the second orderrandom waveform it tends to be Gaussian whereas it is not for theperiodic waves. Single tooth machined surfaces can usually be classifiedsomewhere in between the second order random and the periodic.

Difficulties are experienced when attempting to measure the averagewavelength index of certain profiles, such as square edges which producewaveforms rich in harmonics. It has been found in practice that asignificant value cannot be obtained unless the high frequencycomponents fall off at a rate exceeding 6dB/octave, and it is to ensurethat this condition applies that the low pass filter 7 has been includedin the circuits of the embodiments described.

Filtering is also necessary in practical instruments to remove vibrationand extraneous noise, and from theoretical considerations it has beenfound that the optimum turnover frequency is approximately 50 100 timesthat of the low pass filter and, since the average wavelength index isnot critically dependent on the value of the high frequency turnover abandwidth of 100 is conveniently chosen.

Whilst embodiments of the invention relating to the study of profiles ofsurfaces have been described, it will be apparent to those skilled inthe art that it is not limited to such applications and thatmodifications of the method and apparatus can be applied to theassessment of the relationship between any two interdependentvariables.In general the method will find application where it is desired withsimple instrumentation to obtain a means of comparing the power spectraof sets of inter-related variables.

What is claimed is:

1. Apparatus for characterizing the spectrum of a dependent variablewith respect to an independent variable, said apparatus comprising:

means determining the value of said dependent variable over a range ofvalues of said independent variable,

means providing a first signal representing the mean modulus of saiddependent variable over said range,

means providing a second signal representing the.

mean modulus of the rate of change of said dependent variable withrespect to said independent variable over said range, and

modifying means for modifying one of said first and second signals bymeans of the other of said first and second signals to provide a thirdsignal representing an average wavelength parameter characterizing saidspectrum.

2. The apparatus of claim 1, wherein said means providing said secondsignal representing the mean modulus of said rate of change of saiddependent variable with respect to said independent variable comprises arectifier feeding an integrator and fed with a signal representing therate of change of said dependent variable with respect to saidindependent variable.

3. The apparatus of claim 2 wherein said second signal is obtained atthe output of a differentiation circuit fed with said signalrepresenting said value of said dependent variable.

4. The apparatus of claim 1, wherein said parameter to be determinedcharacterizes the profile of a surface of a body,

said apparatus further comprising means for determining the distance ofsaid surface from a reference line, thereby to provide a signalrepresenting said dependent variable over a range of positions spacedover said surface.

5. The apparatus of claim 4 further comprising a probe movable withrespect to said surface along a line in contact with said surface andresponsive to changes in said profile of said surface to produce asignal in dependence thereon,

said probe comprising said means determining said distance of saidsurface from said reference line.

6. The apparatus of claim 5 wherein said probe is a transducer operableat a modulation frequency, and further comprising:

a carrier oscillator from which said transducer is fed, said carrieroscillator determining the modulation frequency,

means for demodulating said signal from said transducer, and

means for removing the low frequency component of said demodulatedsignal.

7. The apparatus of claim 6, further comprising means for removing anycomponents of said demodulated signal having a frequency greater thanabout 6 octaves above the lowest frequency passed by said means forremoving said low frequency component of said demodulated signal. V

8. The apparatus of claim 5 further comprising means for rectifying andintegrating said demodulated signal,

means for differentiating said demodulated signal,

and

means for rectifying and integrating said differentiated signal toprovide respectively a first signal representing said mean modulus ofthe distance of said surface of said profile from said reference lineand a second signal representing the mean modulus of the rate of changeof said distance of said surface from said reference line. 9. Theapparatus of claim 8 further comprising a divider circuit for producingan output signal representing the division of said first signalrepresenting said mean modulus of the distance of said surface from saidreference line by said second signal representing said mean modulus ofthe rate of change of the distance of said surface from said referenceline.

10. The apparatus of'claim 8 wherein said means for rectifying saiddemodulated signal are full wave rectifiers,

l l. The apparatus of claim 5 further comprising means for rectifyingand integrating said demodulated signal, H

means for generating a plurality of reference signals representing aseries of attainable values of. said demodulated signal,

means for detecting the crossing by said demodulated signal of saidreference value to produce an output pulse in dependence thereon, and

means for counting said pulses produced by said detector, thereby toproduce a numerical representation of the rate of change of saiddistance of said surface from said reference line.

12. The apparatus of claim further comprising:

means for rectifying and integrating said demodulated signal,

means for generating a plurality of reference Signals representing aseries of attainable values of said demodulated signal,

means for detecting the crossing by said demodulated signal of saidreference value to produce an output pulse in dependence thereon, and

means for counting said pulses produced by said detector to produce anumerical representation of the rate of change of said distance of saidsurface from said surface line. 13. The apparatus of claim 4 comprisingmeans for displaying said first and second signals respectivelyrepresenting said mean modulus of said distance of said surface fromsaid reference line, said mean modulus of said rate of change of saiddistance of said surface from said reference line, and said signalrepresenting said parameter determined by said combination of said twosignals.

14. Apparatus according to claim 1 wherein said modifying meanscomprises means for modifying the first signal by means of the secondsignal to provide the third signal, said third signal directlyrepresenting said average wavelength parameter.

15. The apparatus of claim 14 wherein said modifying means modifyingsaid first signal by means of said second signal is a divider circuit inwhich said first signal is divisible by said second signal.

16. Apparatus according to claim 1 wherein said modifying meanscomprises means for modifying the second signal by means of the firstsignal to provide the third signal, the third signal representing thereciprocal of said average wa'velength parameter.

17. The apparatus of claim 16 wherein said modifying means modifyingsaid second signal by means of said first signal is a divider circuit inwhich said second signal is divisible by said first signal.

18. Apparatus for characterizing the spectrum of a dependent variablewith respect to an independent variable, said apparatus comprising:

means determining the value of said dependent variable over a range ofvalues of said independent variable,

means providing a first signal representing the root mean square of saiddependent variable over said range,

means providing a second signal representing the root mean square of therate of change of said dependent variable with respect to saidindependent variable over said range, and

modifying means for modifying one of said first and second signals bymeans of the other of said first and second signals to provide a thirdsignal representing a roof mean square wavelength parametercharacterizing said spectrum.

19. Apparatus according to claim 18 wherein said modifying meanscomprises means for modifying the first signal by means of the secondsignal to provide the third signal, said third signal directlyrepresenting said root mean square wavelength parameter.

20. The apparatus of claim 19 wherein said modifying means modifyingsaid first signal by means of said second signal is a divider circuit inwhich said first signal is divisible by said second signal.

21. Apparatus according to claim 18 wherein said modifying meanscomprises means for modifying the second signal by means of the firstsignal to provide the third signal, the third signal representing thereciprocal of said root mean square wavelength parameter.

22. The apparatus of claim 11 wherein said modifying means modifyingsaid second signal by means of said first signal is a divider circuit inwhich said second signal is divisible by said first signal.

23. The apparatus of claim 18 wherein said means providing said secondsignal representing the root mean square of said rate of change ofsaiddependent variable with respect to said independent variable comprises arectifier feeding an integrator and fed from a differentiationxcircuitwith a signal representing the rate of change of said dependent variablewith respect to said independent variable.

24. The apparatus of claim 23 wherein said signal representing said rateof change of said dependent variable with respect to said independentvariable is obtained at the output of the differentiation circuit fedwith said signal representing said value of said dependent variable. v e

25. The apparatus of claim 18 wherein said parameter to be determinedcharacterizes the profile of a surface of a body said apparatusincluding means for determining the distance of said surface from areference line, thereby to provide a signal representing said dependentvariable over a range of positions spacedover said surface.

26. The apparatus of claim 25 further comprising:

a probe movable with respect to said surface along a line in contactwith said surface and responsive to changes in said profile of saidsurface to produce a signal in dependence thereon,

said probe comprising said means determiningsaid distance of saidsurface from said reference line.

27. The apparatus of claim 26 wherein said probe is a transduceroperable at a modulation frequency, and further comprising:

a carrier oscillator from which said transducer is fed, said carrieroscillator determining said modulation frequency,

means for demodulating said signal from said transducer, and

means for removing any low frequency component of said demodulatedsignal.

28. The apparatus of claim 27, further comprising means for removing thecomponents of said demodulated signal having a frequency greater thanabout 6 octaves above the lowest frequency passed by said means forremoving said low frequency component of said demodulated signal.

29. The apparatus of claim 26 further comprising:

means for rectifying and integrating said demodulated signal,

means for differentiating said demodulated signal,

and

means for rectifying and integrating said differentiated signal toprovide respectively a first signal representing said root mean squareof the distance of said surface of said profile from said reference lineand a second signal representing the root mean square of the rate ofchange of said distance of said surface from said reference line.

30. The apparatus of claim 29 further comprising a divider circuit forproducing an output signal representing the division of said firstsignal representing said root mean square of the distance of saidsurface from said reference line by said second signal representing saidroot mean square of the rate of change of the distance of said surfacefrom said reference line.

31. The apparatus of claim 29 wherein said means for rectifying saiddemodulated signal are square law rectifiers.

32. The apparatus of claim 25 further comprising means for displayingsaid first and second signals respectively representing said root meansquare of said distance of said surface from said reference line, saidroot mean square of said rate of change of said distance of said surfacefrom said reference line, and said signal representing said parameterdetermined by said combination of said two signals.

33. A method of characterizing the spectrum of a dependent variable withrespect to an independent variable comprising the steps of:

determining the value of said dependent variable over a range of valuesof said independent variable, deriving a first signal representing themean modulus of said dependent variable over said range,

deriving a second signal representing the mean modulus of the rate ofchange of said dependent variable with respect to said independentvariable over said range, and

modifying one of said first and second signals by means of the other ofsaid first and second signals to provide a third signal representing anaverage wavelength parameter characterizing said spectrum.

34. The method of claim 33 wherein said first signal representing saidmean modulus of the rate of change of said dependent variable over saidrange is derived from a signal representing the value of said rate ofchange of said dependent variable which in turn is derived bydifferentiation of a signal representing the value of said dependentvariable over said range.

35. The method of claim 33 wherein said parameter to be determinedcharacterizes the profile of a surface of a body, said dependentvariable being the distance of said surface of said body from areference line and said independent variable being distance along saidsurface from a predetermined point.

36. The method of claim 35 wherein said signal representing the value ofsaid dependent variable is obtained by passing a probe in contact withsaid surface along a predetermined line to obtain a signal representingthe distance of said surface from a reference line.

37. The method of claim 36 wherein a signal representing said rate ofchange of said dependent variable is obtained by differentiation of saidsignal representing the distance of said surface from said referenceline and is used to provide said first signal representing said meanmodulus of said rate of change of said distance of said surface fromsaid reference line.

38. The method of claim 35, wherein said parameter determined is therate of said mean modulus of distance of said surface from saidreference line to said mean modulus of said rate of change of saiddistance of said surface from said reference line.

39. A method according to claim 33 wherein the step i of modifyingcomprises modifying the first signal by means of the second signal toprovide the third signal, said third signal directly representing saidaverage wavelength parameter.

40. The method of claim 39 wherein said first signal is divided by saidsecond signal to provide said third signal representing the averagewavelength parameter characterizing said spectrum.

4l. A method according to claim 33 wherein the step of modifyingcomprises modifying the second signal by means of the first signal toprovide the third signal, the third signal representing the reciprocalof said average wavelength parameter.

42. The method of claim 41 wherein said second signal is divided by saidfirst signal to provide said signal representing the reciprocal of theaverage wavelength parameter characterizing said spectrum.

43. A method of characterizing the spectrum of a dependent variable withrespect to an independent variable comprising the steps of:

determining the value of said dependent variable over a range of valuesof said independent variable, deriving a first signal representing theroot mean square of said dependent variable over said range,

deriving a second signal representing the root mean square of the rateof change of said dependent variable with respect to said independentvariable over said range, and

modifying one of said first and second signals by means of the other ofsaidfirst and second signals to provide a third signal representing aroot mean square wavelength parameter characterizing said spectrum.

44. A method according to claim 43 wherein the step of modifyingcomprises modifying the first signal by means of the second signal toprovide the third signal, said third signal directly representing saidroot mean square wavelength parameter.

45. The method of claim 44 wherein said first signal is divided by saidsecond signal to provide said third signal representing the root meansquare wavelength parameter characterizing said spectrum.

46. A method according to claim 43 wherein the step of modifyingcomprises modifying the second signal by means of the first signal toprovide the third signalrthe third signal representing the reciprocal ofsaid root mean square parameter wavelength.

47. The method of claim 46 wherein said second signal is divided by saidfirst signal to provide said third signal representing the reciprocalof. the root mean square wavelength parameter characterizing saidspectrum.

48. The method of claim 43 wherein said first signal representing saidroot mean square of the rate of change of said dependent variable oversaid range is derived from a signal representing the value of said rateof change of said dependent variable which is derived by differentiationof a signal representing the value of said dependent variable over saidrange.

49. The method of claim 43 wherein said parameter to be determinedcharacterizes the profile of a surface of a body, said dependentvariable being the distance of said surface of said body from areference line and said independent variable being distance along saidsurface from a predetermined point.

50. The method of claim 49 wherein said signal representing the value ofsaid dependent variable is obtained by passing a probe in contact withsaid surface along a predetermined line to obtain a signal representingthe difference of said surface from a reference line.

51. The method of claim 50 wherein a signal representing said rate ofchange of said dependent variable is obtained by differentiation of saidsignal representing the distance of said surface from said referenceline and used to provide said first signal representing said root meansquare of said rate of change of said distance of said surface from saidreference line.

52. The method of claim 49 wherein said parameter determined is theratio of said root mean square of distance of said surface from saidreference line to said root mean square of said rate of change of saiddistance of said surface from said reference line.

1. Apparatus for characterizing the spectrum of a dependent variablewith respect to an independent variable, said apparatus comprising:means determining the value of said dependent variable over a range ofvalues of said independent variable, means providing a first signalrepresenting the mean modulus of said dependent variable over saidrange, means providing a second signal representing the mean modulus ofthe rate of change of said dependent variable with respect to saidindependent variable over said range, and modifying means for modifyingone of said first and second signals by means of the other of said firstand second signals to provide a third signal representing an averagewavelength parameter characterizing said spectrum.
 1. Apparatus forcharacterizing the spectrum of a dependent variable with respect to anindependent variable, said apparatus comprising: means determining thevalue of said dependent variable over a range of values of saidindependent variable, means providing a first signal representing themean modulus of said dependent variable over said range, means providinga second signal representing the mean modulus of the rate of change ofsaid dependent variable with respect to said independent variable oversaid range, and modifying means for modifying one of said first andsecond signals by means of the other of said first and second signals toprovide a third signal representing an average wavelength parametercharacterizing said spectrum.
 2. The apparatus of claim 1, wherein saidmeans providing said second signal representing the mean modulus of saidrate of change of said dependent variable with respect to saidindependent variable comprises a rectifier feeding an integrator and fedwith a signal representing the rate of change of said dependent variablewith respect to said independent variable.
 3. The apparatus of claim 2wherein said second signal is obtained at the output of adifferentiation circuit fed with said signal representing said value ofsaid dependent variable.
 4. The apparatus of claim 1, wherein saidparameter to be determined characterizes the profile of a surface of abody, said apparatus further comprising means for determining thedistance of said surface from a reference line, thereby to provide asignal representing said dependent variable over a range of positionsspaced over said surface.
 5. The apparatus of claim 4 further comprisinga probe movable with respect to said surface along a line in contactwith said surface and responsive to changes in said profile of saidsurface to produce a signal in dependence thereon, said probe comprisingsaid means determining said distance of said surface from said referenceline.
 6. The apparatus of claim 5 wherein said probe is a transduceroperable at a modulation frequency, and further comprising: a carrieroscillator from which said transducer is fed, said carrier oscillatordetermining the modulation frequency, means for demodulating said signalfrom said transducer, and means for removing the low frequency componentof said demodulated signal.
 7. The apparatus Of claim 6, furthercomprising means for removing any components of said demodulated signalhaving a frequency greater than about 6 octaves above the lowestfrequency passed by said means for removing said low frequency componentof said demodulated signal.
 8. The apparatus of claim 5 furthercomprising means for rectifying and integrating said demodulated signal,means for differentiating said demodulated signal, and means forrectifying and integrating said differentiated signal to providerespectively a first signal representing said mean modulus of thedistance of said surface of said profile from said reference line and asecond signal representing the mean modulus of the rate of change ofsaid distance of said surface from said reference line.
 9. The apparatusof claim 8 further comprising a divider circuit for producing an outputsignal representing the division of said first signal representing saidmean modulus of the distance of said surface from said reference line bysaid second signal representing said mean modulus of the rate of changeof the distance of said surface from said reference line.
 10. Theapparatus of claim 8 wherein said means for rectifying said demodulatedsignal are full wave rectifiers.
 11. The apparatus of claim 5 furthercomprising means for rectifying and integrating said demodulated signal,means for generating a plurality of reference signals representing aseries of attainable values of said demodulated signal, means fordetecting the crossing by said demodulated signal of said referencevalue to produce an output pulse in dependence thereon, and means forcounting said pulses produced by said detector, thereby to produce anumerical representation of the rate of change of said distance of saidsurface from said reference line.
 12. The apparatus of claim 5 furthercomprising: means for rectifying and integrating said demodulatedsignal, means for generating a plurality of reference signalsrepresenting a series of attainable values of said demodulated signal,means for detecting the crossing by said demodulated signal of saidreference value to produce an output pulse in dependence thereon, andmeans for counting said pulses produced by said detector to produce anumerical representation of the rate of change of said distance of saidsurface from said surface line.
 13. The apparatus of claim 4 comprisingmeans for displaying said first and second signals respectivelyrepresenting said mean modulus of said distance of said surface fromsaid reference line, said mean modulus of said rate of change of saiddistance of said surface from said reference line, and said signalrepresenting said parameter determined by said combination of said twosignals.
 14. Apparatus according to claim 1 wherein said modifying meanscomprises means for modifying the first signal by means of the secondsignal to provide the third signal, said third signal directlyrepresenting said average wavelength parameter.
 15. The apparatus ofclaim 14 wherein said modifying means modifying said first signal bymeans of said second signal is a divider circuit in which said firstsignal is divisible by said second signal.
 16. Apparatus according toclaim 1 wherein said modifying means comprises means for modifying thesecond signal by means of the first signal to provide the third signal,the third signal representing the reciprocal of said average wavelengthparameter.
 17. The apparatus of claim 16 wherein said modifying meansmodifying said second signal by means of said first signal is a dividercircuit in which said second signal is divisible by said first signal.18. Apparatus for characterizing the spectrum of a dependent variablewith respect to an independent variable, said apparatus comprising:means determining the value of said dependent variable over a range ofvalues of said independent variable, means providing a first signalrepresenting the root mean squarE of said dependent variable over saidrange, means providing a second signal representing the root mean squareof the rate of change of said dependent variable with respect to saidindependent variable over said range, and modifying means for modifyingone of said first and second signals by means of the other of said firstand second signals to provide a third signal representing a roof meansquare wavelength parameter characterizing said spectrum.
 19. Apparatusaccording to claim 18 wherein said modifying means comprises means formodifying the first signal by means of the second signal to provide thethird signal, said third signal directly representing said root meansquare wavelength parameter.
 20. The apparatus of claim 19 wherein saidmodifying means modifying said first signal by means of said secondsignal is a divider circuit in which said first signal is divisible bysaid second signal.
 21. Apparatus according to claim 18 wherein saidmodifying means comprises means for modifying the second signal by meansof the first signal to provide the third signal, the third signalrepresenting the reciprocal of said root mean square wavelengthparameter.
 22. The apparatus of claim 11 wherein said modifying meansmodifying said second signal by means of said first signal is a dividercircuit in which said second signal is divisible by said first signal.23. The apparatus of claim 18 wherein said means providing said secondsignal representing the root mean square of said rate of change of saiddependent variable with respect to said independent variable comprises arectifier feeding an integrator and fed from a differentiation circuitwith a signal representing the rate of change of said dependent variablewith respect to said independent variable.
 24. The apparatus of claim 23wherein said signal representing said rate of change of said dependentvariable with respect to said independent variable is obtained at theoutput of the differentiation circuit fed with said signal representingsaid value of said dependent variable.
 25. The apparatus of claim 18wherein said parameter to be determined characterizes the profile of asurface of a body said apparatus including means for determining thedistance of said surface from a reference line, thereby to provide asignal representing said dependent variable over a range of positionsspaced over said surface.
 26. The apparatus of claim 25 furthercomprising: a probe movable with respect to said surface along a line incontact with said surface and responsive to changes in said profile ofsaid surface to produce a signal in dependence thereon, said probecomprising said means determining said distance of said surface fromsaid reference line.
 27. The apparatus of claim 26 wherein said probe isa transducer operable at a modulation frequency, and further comprising:a carrier oscillator from which said transducer is fed, said carrieroscillator determining said modulation frequency, means for demodulatingsaid signal from said transducer, and means for removing any lowfrequency component of said demodulated signal.
 28. The apparatus ofclaim 27, further comprising means for removing the components of saiddemodulated signal having a frequency greater than about 6 octaves abovethe lowest frequency passed by said means for removing said lowfrequency component of said demodulated signal.
 29. The apparatus ofclaim 26 further comprising: means for rectifying and integrating saiddemodulated signal, means for differentiating said demodulated signal,and means for rectifying and integrating said differentiated signal toprovide respectively a first signal representing said root mean squareof the distance of said surface of said profile from said reference lineand a second signal representing the root mean square of the rate ofchange of said distance of said surface from said reference line. 30.The apparatus of claim 29 further cOmprising a divider circuit forproducing an output signal representing the division of said firstsignal representing said root mean square of the distance of saidsurface from said reference line by said second signal representing saidroot mean square of the rate of change of the distance of said surfacefrom said reference line.
 31. The apparatus of claim 29 wherein saidmeans for rectifying said demodulated signal are square law rectifiers.32. The apparatus of claim 25 further comprising means for displayingsaid first and second signals respectively representing said root meansquare of said distance of said surface from said reference line, saidroot mean square of said rate of change of said distance of said surfacefrom said reference line, and said signal representing said parameterdetermined by said combination of said two signals.
 33. A method ofcharacterizing the spectrum of a dependent variable with respect to anindependent variable comprising the steps of: determining the value ofsaid dependent variable over a range of values of said independentvariable, deriving a first signal representing the mean modulus of saiddependent variable over said range, deriving a second signalrepresenting the mean modulus of the rate of change of said dependentvariable with respect to said independent variable over said range, andmodifying one of said first and second signals by means of the other ofsaid first and second signals to provide a third signal representing anaverage wavelength parameter characterizing said spectrum.
 34. Themethod of claim 33 wherein said first signal representing said meanmodulus of the rate of change of said dependent variable over said rangeis derived from a signal representing the value of said rate of changeof said dependent variable which in turn is derived by differentiationof a signal representing the value of said dependent variable over saidrange.
 35. The method of claim 33 wherein said parameter to bedetermined characterizes the profile of a surface of a body, saiddependent variable being the distance of said surface of said body froma reference line and said independent variable being distance along saidsurface from a predetermined point.
 36. The method of claim 35 whereinsaid signal representing the value of said dependent variable isobtained by passing a probe in contact with said surface along apredetermined line to obtain a signal representing the distance of saidsurface from a reference line.
 37. The method of claim 36 wherein asignal representing said rate of change of said dependent variable isobtained by differentiation of said signal representing the distance ofsaid surface from said reference line and is used to provide said firstsignal representing said mean modulus of said rate of change of saiddistance of said surface from said reference line.
 38. The method ofclaim 35, wherein said parameter determined is the rate of said meanmodulus of distance of said surface from said reference line to saidmean modulus of said rate of change of said distance of said surfacefrom said reference line.
 39. A method according to claim 33 wherein thestep of modifying comprises modifying the first signal by means of thesecond signal to provide the third signal, said third signal directlyrepresenting said average wavelength parameter.
 40. The method of claim39 wherein said first signal is divided by said second signal to providesaid third signal representing the average wavelength parametercharacterizing said spectrum.
 41. A method according to claim 33 whereinthe step of modifying comprises modifying the second signal by means ofthe first signal to provide the third signal, the third signalrepresenting the reciprocal of said average wavelength parameter. 42.The method of claim 41 wherein said second signal is divided by saidfirst signal to provide said signal representing the reciprocal of theaverage wavelength parameter characterizing said spectrum.
 43. A methodof characterizing the spectrum of a dependent variable with respect toan independent variable comprising the steps of: determining the valueof said dependent variable over a range of values of said independentvariable, deriving a first signal representing the root mean square ofsaid dependent variable over said range, deriving a second signalrepresenting the root mean square of the rate of change of saiddependent variable with respect to said independent variable over saidrange, and modifying one of said first and second signals by means ofthe other of said first and second signals to provide a third signalrepresenting a root mean square wavelength parameter characterizing saidspectrum.
 44. A method according to claim 43 wherein the step ofmodifying comprises modifying the first signal by means of the secondsignal to provide the third signal, said third signal directlyrepresenting said root mean square wavelength parameter.
 45. The methodof claim 44 wherein said first signal is divided by said second signalto provide said third signal representing the root mean squarewavelength parameter characterizing said spectrum.
 46. A methodaccording to claim 43 wherein the step of modifying comprises modifyingthe second signal by means of the first signal to provide the thirdsignal, the third signal representing the reciprocal of said root meansquare parameter wavelength.
 47. The method of claim 46 wherein saidsecond signal is divided by said first signal to provide said thirdsignal representing the reciprocal of the root mean square wavelengthparameter characterizing said spectrum.
 48. The method of claim 43wherein said first signal representing said root mean square of the rateof change of said dependent variable over said range is derived from asignal representing the value of said rate of change of said dependentvariable which is derived by differentiation of a signal representingthe value of said dependent variable over said range.
 49. The method ofclaim 43 wherein said parameter to be determined characterizes theprofile of a surface of a body, said dependent variable being thedistance of said surface of said body from a reference line and saidindependent variable being distance along said surface from apredetermined point.
 50. The method of claim 49 wherein said signalrepresenting the value of said dependent variable is obtained by passinga probe in contact with said surface along a predetermined line toobtain a signal representing the difference of said surface from areference line.
 51. The method of claim 50 wherein a signal representingsaid rate of change of said dependent variable is obtained bydifferentiation of said signal representing the distance of said surfacefrom said reference line and used to provide said first signalrepresenting said root mean square of said rate of change of saiddistance of said surface from said reference line.